1.01.2005

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» Q&A: What's the cashflow if an equity swap...
» Q&A: What's the cutoff for the bottom 10% of an exam...
» Q&A: What's the Delta of a call option on...
» Q&A: What's the difference between an Interest Rate Option and...
» Q&A: What's the difference between discretely and continuously compounded rates of return?
» Q&A: What's the end-of-day P&L if an investor...
» Q&A: What's the expected number of correct answers...
» Q&A: What's the expiration-day P&L for the protective put...
» Q&A: What's the expiration-day P&L of the covered call position if...
» Q&A: What's the maximum profit for the covered call...

Q&A: What's the cashflow if an equity swap...

...has been arranged between an investor and a dealer. The notional principal is $100 million, and payments will be made semiannually. The investor agrees to pay the dealer the return on a small-cap stock index, and the dealer agrees to pay a fixed rate of 6.75 percent, with payments made on the basis of 182 days in the period and 365 days in a year. The small-cap index starts off at 1,805.20; six months later it is at 1,796.15.

Answer

  • The fixed payment is $100,000,000 x 0.0675 x 182/365 = $3,365,753.
  • The equity payment is (1796.15/1805.20 - 1) x $100,000,000 = -$501,329.
  • The cashflow from the dealer to the investor is $3,365,753 + $501,329 = $3,867,082.

Category: C++ Quant > Finance > Derivatives > Swaps

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Q&A: What's the cutoff for the bottom 10% of an exam...

...with scores normally distributed with m = 80 and s = 5?

Answer : P(X<73.6) = 10%

  • In Excel: z = NORMSINV(10%) = -1.28.
    • A z-score of -1.28 means that the original score/observation is 1.28 standard deviations below, not above, the mean.
  • Since z = (X - m)/s, X = z*s + m = -1.28*5 + 80 = 73.6.

Category: C++ Quant > Finance > Quantitative Analysis > Probability

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Q&A: What's the Delta of a call option on...

...a stock trading at $100. The continuously compounding risk-free interest rate is 6%. The option has an exercise price of $100 and expires in one year. The standard deviation of the stock's returns is 0.1.

Answer : Delta = change in option price / change in underlying price, which can be approximated from the Black-Scholes-Merton formula as the value of N(d1) for calls and N(d1) - 1 for puts.

  • d1 = ( ln(S0/X) + (r+stderr^2/2)*T) / (stderr*T^1/2) = ( ln(100/100) + (0.06 + 0.12/2)*1) / (0.1*1^(1/2)) = 0.65
  • N(d1): In Excel, Normsdist(0.65) = 0.74
  • For a $1 up(+) in the price of the stock, we expect the new call option price to be up(+) by 0.74.

Bonus Points

  • The delta of a call option is always positive as the value of a call increases with an underlying price increase.
  • The delta of a put option is always negative as the value of a put decreases with an underlying price increase.
    • The put delta = N(d1) - 1 = 0.74 - 1 = -.026.
    • For a $1 up (+) in the price of the stock, we expect the new put option price to be down(-) by 0.26.
  • In general, the larger the move in the underlying price, the worse the approximation.

Category: C++ Quant > Finance > Derivatives > Options

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Q&A: What's the difference between an Interest Rate Option and...

...a Forward Rate Agreement?

Answer : the underlyings are both interest rate, and both have notional principals, but a FRA is a commitment to make one interest payment and receive another, whereas an interest rate options is a right.

Category: C++ Quant > Finance > Derivatives > Options

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Q&A: What's the difference between discretely and continuously compounded rates of return?

Answer : A discretely compounded rate of return measures the rate of changes in the value of asset over a period under the assumption that the number of compounding periods is countable. Most standard deposit and loan instruments are compounded at discrete and evenly spaced periods, such as annually or monthly. For example, suppose that the holding period return on a stock over a year is 50%. If the rate of return is compounded on a quarterly basis, the compounded quarterly rate of return on the stock is ( 1 + 0.5)^(1/4) - 1 = 10.67%.

The continuously compounded rate of return assumes continuously compounding. It is the natural logarithm of 1 plus the holding period return, or equivalently, the natural logarithm of the ending price over the beginning price. From t to t+1: r(t, t+1) = ln( S(t+1)/S(t) ) = ln(1 + R(t, t+1)), where S is stock price.

Category: C++ Quant > Finance > Valuation

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Q&A: What's the end-of-day P&L if an investor...

...purchases 2 Treasury futures contracts at 103 and the contracts are marked to the market at a price of 100?

The investor has to pay the clearing house 2 * (103,000 - 100,000) = $6,000. (Marked to market means contracts are adjusted daily to reflect the price changes and Investors are liable for gains or losses.)

Category: C++ Quant > Finance > Derivatives > Futures

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Q&A: What's the expected number of correct answers...

in this interview exam?

Answer : mean = np = 5 x 0.25 = 1.25, with a standard deviation of = (n*p*(1 - p)) ^0.5 = (5 * 0.25 * 0.75)^0.5 = 0.47.

  • can NOT approximate the binomial with a normal distribution: approximation works best if np >= 5 and n(1-p) >= 5.

Bonus Points

  • The normal approximation to the binomial is useful whenever the binomial tables cannot be used (ie. the number of trials is so large).
  • Since the normal distribution is symmetrical, it best approximates binomial distributions that are reasonably symmetrical (ie. p close to 0.5).

Category: C++ Quant > Finance > Probability > Distribution

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Q&A: What's the expiration-day P&L for the protective put...

...if an investor purchases 100 shares of a stock for $50/share, buys put options on that stock with an exercise price of $38 for a premium of $3/share, and the expiration-day price is $55?

Answer : Value of protected put = value of underlying + value of put call = ($55-50-3) * 100 = $200.

Bonus Points

  • Protective Put = Stock + a Long Put
  • Investors would likely buy a protective put (portfolio insurance) if they are concerned about a downward price movement in the stock.

Category: C++ Quant > Finance > Derivatives > Options

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Q&A: What's the expiration-day P&L of the covered call position if...

...an investor buys a share of stock for $20, and simultaneously sells a call option on that stock for $5? The exercise price of the call is $30, the call will expire in 3 months, and the stock finishes at $18? What if the stock price finishes at $32? $40?

Answer :

  • the portfolio costs $20 - 5 = $15
  • if stock price finishes at $18: the value of the covered call = $18 (the value of the stock) + $5 (the option premium) = $23. P&L = 23 - 15 = $8
  • if $32: the buyer will exercise the option = $30 + $5 = $35. P&L = 35 - 15 = $20
  • $40: same as above.

Bonus Points

  • Value of covered call = value of underlying + value of short call
  • Valueof the underlying = MAX(0, ST - X) = ST if ST <= X, or X if ST > X.

Category: C++ Quant > Finance > Derivatives > Options

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Q&A: What's the maximum profit for the covered call...

...if an investor purchases 100 shares of a stock for $38/share, sells call options on that stock with an exercise price of $50 for a premium of $3/share, and holds the option to expiration?

Answer : Value of covered call = value of underlying + value of short call = ($50-38+3) * 100 = $1500.

Bonus Points

  • Covered Call = Stock + a Short Call. In a covered call transactions, a trader is generally assumed to already own the underlying (the term "covered" means that the potential obligation to deliver the underlying is covered by the underlying.)
  • Invetors would likely write a covered call if they do not anticipate a significant change in stock price in the near term.

Category: C++ Quant > Finance > Derivatives > Options

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